Diffusion-controlled growth rate of stepped interfaces

For many materials, the structure of crystalline surfaces or solid-solid interphase boundaries is characterized by an array of mobile steps separated by immobile terraces. Despite the prevalence of step-terraced interfaces a theoretical description of the growth rate has not been completely solved. In this work the boundary element method (BEM) has been utilized to numerically compute the concentration profile in a fluid phase in contact with an infinite array of equally spaced surface steps and, under the assumption that step motion is controlled by diffusion through the fluid phase, the growth rate is computed. It is also assumed that a boundary layer exists between the growing surface and a point in the liquid where complete convective mixing occurs. The BEM results are presented for varying step spacing, supersaturation, and boundary layer width. BEM calculations were also used to study the phenomenon of step bunching during crystal growth, and it is found that, in the absence of elastic strain energy, a sufficiently large perturbation in the position of a step from its regular spacing will lead to a step bunching instability. Finally, an approximate analytic solution using a matched asymptotic expansion technique is presented for the case of a stagnant liquid or equivalently a solid-solid stepped interface.